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# An abstract algebra problem!

What is the strongest condition you can up with for $$\alpha$$ such that the field extension over $$F$$ is equal to the ring of polynomials over $$F$$. Or in other words $$F(\alpha)=F[\alpha]$$?

Note by Ali Caglayan
3 years, 5 months ago

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The strongest I could come up with is that $$\alpha$$ generates a group such that $$\langle\alpha\rangle\cong C_n$$ where $$C_n$$ is the $$n^\text{th}$$ cyclic group of arbitrary order.

- 3 years, 5 months ago

I have gone further and said that $$\alpha$$ has to be algebraic over $$F$$.

- 3 years, 5 months ago